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Fantasia in D-Minor from Mozart K397 is written in 2/2 (alla breve). The first part of it should be played Andante (let's say 80 half notes in one minute). In reality, it is played much much slower and it is more played as it would be 4/4.

The same is to some extent true for the first movement of the Moonlight sonata, and maybe there are other examples that I'm currently not aware of.

So what explains the difference in the tempo indicated and the tempo used by most pianists? Why are the pieces more played as it would be 4/4 instead of 2/2?

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In Right Before Your Eyes: A Fresh Approach to Interpreting a Piano Score, Ruth Price reasons at length about the tempo of this movement (Chapter 6). It's extremely personal and subjective and makes no attempt at scholarship (it needs to "feel meditative"), but it makes a good point that sometimes specific musical qualities influence the choice of tempo.

A more scholarly viewpoint is Helmut Breidenstein's Mozart's Tempo-System: A Handbook for Practice and Theory. It makes it clear that, for one thing, it gets horribly complicated when figuring out what alla breve is all about, and that "Andante" can mean a lot of things.

Contrary to our habit, the tempo word alone was no "tempo indication in the late 18th century. ... "Andante" in the 18th century does not mean the stride or walking pace of a human being but the incorporeal pace of the music, a "walking motion" in the sense of a regular progression of time.
In 1786 Friedrich Nicolai reports on the significant differences in the performance practice of Andante between Berlin, Dresden, and Vienna. In general, however, the 18th century regarded Andante as "the tempo which keeps the mean between swift and slow." It was only in the 19th century that Andante received the meaning of "slow," which influenced Mozart interpretation disastrously from then on."

Taking both Price's and Breidenstein's points together, that adds up to:

  1. Although there may be significant interpretive practice giving this movement a fairly slow pace, you can definitely use a historical argument to take a faster tempo;
  2. Given the confusion about cut time, you don't have to feel compelled solely by the cut time marking to take a certain tempo; and
  3. There is such a thing as "too fast for the music." Don't hurt anything!
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There is a fallacy in the question: The Mozart D Minor Fantasy is written in common time (i.e., 4/4). This can be confirmed by browsing the various editions on IMSLP.

The primary exception to this (on IMSLP) is a Soviet-era score from Russia, which is given in cut time. However, such scores are notorious for containing errors and should not be considered a reliable source.

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    I'm using a Henle Urtext publication as a reference.... and even then, the question in general (for the Beethoven Sonata) remains valid
    – Rian
    Feb 23 at 0:13
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    @Rian Fair enough. Suggest you edit your question (the title particularly) to focus on the general issue. Also, mention the Henle (including an image, if possible), so you don't get people like me calling the edition into question.
    – Aaron
    Feb 23 at 0:18
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    @Rian BTW, does the Henle include any commentary regarding the time signature?
    – Aaron
    Feb 23 at 0:18
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First, "of course", pianists who feel sufficiently confident to make their own interpretation choices will do so! :)

Incidental to that is what I see as a sort of backlash against aggressively up-tempo interpretations that don't really sound any better, by any means, but are more "virtuosic".

Second, editors of editions do make many changes to original markings, or insert markings where there were none in the original.

Third, though I've not researched this, I vaguely recall some scholar asserting that Beethoven (for example) inserted rather fanciful metronome markings (tending to be too fast), because the metronome was a new, cool thing, but/and he'd never really used one. Plausible.

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